Asymptotic behavior of slowly varying solutions of second-order ordinary binomial differential equations with rapidly varying nonlinearity (Q2416801)

The second-order nonlinear ordinary differential equation \[ y''=\alpha_0p(t)\varphi(y) \] is considered, where \(\alpha_0=\pm 1\), \(p\colon [a,\omega)\to [0,\infty)\) is a continuous function, \(-\infty < a < \omega\le \infty\) and \(\varphi\) is a rapidly varying nonlinearity. Conditions for the existence of slowly varying solutions are derived and their asymptotic representations are determined.